🦕 DinoGeo — Geometry Mastery Workbook

📐 Unit 1: Lines, Angles & Proofs ▼

SUPPLEMENTARY

Angles sum to 180°

COMPLEMENTARY

Angles sum to 90°

VERTICAL ANGLES

Equal (across vertex)

LINEAR PAIR

Adjacent; sum = 180°

📌 KEY THEOREMS

Parallel lines cut by transversal:
• Alternate interior ∠s → Equal
• Corresponding ∠s → Equal
• Co-interior (same-side) → 180°
Exterior angle = sum of 2 non-adj. interior ∠s

⭐ MEMORIZE

Vertical angles are ALWAYS equal
Linear pair ALWAYS sums to 180°
Alternate angles need PARALLEL lines
Exterior angle theorem: ∠ext = ∠A + ∠B

💡 EXAMPLE

Two parallel lines are cut by a transversal. One co-interior angle is 65°. Find the other.

Co-interior angles are supplementary.
Other angle = 180° − 65° = 115°

🔺 Unit 2: Triangles & Congruence/Similarity ▼

ANGLE SUM

∠A + ∠B + ∠C = 180°

PYTHAGOREAN

a² + b² = c²

CONGRUENCE

SSS, SAS, ASA, AAS, HL

SIMILARITY

AA, SAS~, SSS~

📌 AREA & SPECIAL TRIANGLES

Area = ½ × base × height

30-60-90: sides → 1 : √3 : 2
45-45-90: sides → 1 : 1 : √2

Midsegment = ½ × parallel side
Similar ratio k → Area ratio k²

⭐ MEMORIZE

SSA is NOT a congruence postulate (ambiguous case)
CPCTC: Corresponding Parts of Congruent Triangles are Congruent
Isosceles: base angles are equal
Centroid divides median in 2:1 ratio

💡 EXAMPLE

Two similar triangles have sides in ratio 3:5. What is the ratio of their areas?

Area ratio = (3)² : (5)² = 9 : 25

▱ Unit 3: Quadrilaterals & Polygons ▼

PARALLELOGRAM

Opp sides ∥ & equal

RECTANGLE

Parallelogram + 90° angles

RHOMBUS

All sides equal; ⊥ diags

SQUARE

Rectangle + Rhombus

📌 POLYGON FORMULAS

Interior angle sum = (n − 2) × 180°
Each interior angle (regular) = (n−2)×180° / n
Exterior angle sum = ALWAYS 360°
Each exterior angle (regular) = 360° / n

Parallelogram area = base × height
Trapezoid area = ½(b₁ + b₂) × h

⭐ MEMORIZE

Parallelogram: diagonals BISECT each other
Rectangle: diagonals are EQUAL length
Rhombus: diagonals are PERPENDICULAR
Square: diagonals bisect, equal, perpendicular
Trapezoid: exactly one pair of parallel sides

💡 EXAMPLE

Find the sum of interior angles of a hexagon.

(n − 2) × 180° = (6 − 2) × 180° = 4 × 180° = 720°

⭕ Unit 4: Circles ▼

CIRCUMFERENCE

C = 2πr = πd

AREA

A = πr²

ARC LENGTH

(θ/360)·2πr

SECTOR AREA

(θ/360)·πr²

📌 CIRCLE ANGLE THEOREMS

Central angle = Intercepted arc
Inscribed angle = ½ × Intercepted arc
Tangent-chord angle = ½ arc
Two chords (inside) = ½(arc₁ + arc₂)
Two secants (outside) = ½|arc₁ − arc₂|
Tangent² = external × whole secant
Chord-chord: AE × EB = CE × ED

⭐ MEMORIZE

Inscribed angle in semicircle = 90°
Tangent ⊥ radius at point of tangency
Equal chords are equidistant from center
Inscribed angles subtending same arc are equal

💡 EXAMPLE

A central angle is 80°. What is the inscribed angle that intercepts the same arc?

Inscribed angle = ½ × central angle = ½ × 80° = 40°

📦 Unit 5: 3D Figures & Coordinate Geometry ▼

📌 VOLUME & SURFACE AREA

Prism: V = Bh, SA = 2B + Ph
Cylinder: V = πr²h, SA = 2πr²+2πrh
Pyramid: V = ⅓Bh
Cone: V = ⅓πr²h, SA = πr²+πrl
Sphere: V = ⁴⁄₃πr³, SA = 4πr²

Distance = √[(x₂−x₁)²+(y₂−y₁)²]
Midpoint = ((x₁+x₂)/2,(y₁+y₂)/2)
Slope = (y₂−y₁)/(x₂−x₁)
Perpendicular slopes: m₁ × m₂ = −1

⭐ MEMORIZE

Cone & Pyramid volume = ⅓ × (prism/cylinder volume)
Sphere SA = 4πr² (4 great circles)
l in cone formula = slant height = √(r² + h²)
Scale factor k → Volume scales by k³

💡 EXAMPLE

A cone has radius 3 cm and height 4 cm. Find its volume.

V = ⅓πr²h = ⅓ × π × 9 × 4 = 12π ≈ 37.7 cm³